Learn what terminating and repeating decimals are and how to transform these rational number from decimal to fractional form.

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The Quick and also Dirty To transform decimals into fractions, simply take what each decimal number represents the end of 1000 and also then include those numbers together. The decimal 0.125 is equal to the sum of 100/1000 + 20/1000 + 5/1000 because that a grand total of (100 + 20 + 5)/1000 = 125/1000.


The number we usage in our day-to-day lives have the right to be damaged up into two key groups: rational and also irrational numbers. Irrational numbers cannot be written out specifically in decimal form since you’d need an infinite number of decimal number to execute so. Rational numbers deserve to be written as decimal numbers that either protect against after some variety of digits or save repeating some pattern of digits forever.In today’s article, we’re going come learn exactly how to take it a decimal representation of a rational number and turn it into an indistinguishable fraction.

What are Terminating and Repeating Decimals?

Before we obtain into the details of how to actually convert terminating and also repeating decimals into fractions, we’d much better make sure we know what it way for a reasonable number to it is in a “terminating” or “repeating” decimal in the an initial place. To watch what the difference is, let’s take a look at a few examples that decimal representations of reasonable numbers:
1/4 = 0.25 is a terminating decimal since it has a finite variety of decimal digits1/3 = 0.3333… is a repeating decimal because the number 3 go on forever3/5 = 0.6 is one more terminating decimal number7/9 = 0.7777… is a repeating decimal since 7 goes on forever9/11 = 0.818181… is another repeating decimal since the sample of digits “81” repeat forever.So a repeating decimal is a rational number whose decimal representation has some repeating pattern, and also a terminating decimal is a rational number whose decimal representation eventually stops. (Remember, a decimal that just goes on and also on through no repeating sample is irrational.)

Can a end Decimal Be composed as a Repeating Decimal?

If friend think about it though, you’ll see that any type of terminating decimal number can actually be composed as a repeating decimal too. How? Well, due to the fact that you can always attach an infinite number of zeros to the an extremely end the a number without transforming its value, you have the right to put one infinitely long string that zeros ~ above the finish of an otherwise end decimal…and you’ll have turned it into a repeating decimal!For example, you have the right to think the the end decimal 0.25 together 0.25000… instead. Yet in this case, no one of this really matters because the worth of the number is specifically the exact same no matter just how it’s written. And also that’s why usually as soon as we speak “repeating decimal,” we median a decimal number wherein something other than just zeros space doing the repeating!

How to convert Decimals come Fractions

Now the we understand the lingo and can phone call the difference between a terminating and repeating decimal, let’s number out exactly how to convert them into fractions. In various other words, in the examples we gave earlier, we stated things prefer “the portion 1/4 is equal to the end decimal 0.25” and also “the portion 7/9 is same to the repeating decimal 0.7777…,” and so on. Yet now let’s figure out exactly how to do this trouble backward so that we can take a decimal number, favor 0.818181…, and also convert it right into a portion with an identical value.

How to convert solitary Digit decimals to Fractions

Let’s start by converting a straightforward terminating decimal number prefer 0.5 into a fraction. As we learned ago in the short article called “What are Decimals?”, a decimal number favor 0.5 way “five the the portion one-tenth.” Which, that course, is simply equal to the fraction 5/10. And also believe it or not, that’s the answer come the problem! So, the decimal 0.5 is identical to the fraction 5/10. Easy, right?Well, it transforms out the we deserve to actually execute a bit more with this fraction. We’ll talk about this in a future article, yet this portion can it is in rewritten so the it’s what’s called “reduced to shortest terms.” without going right into too lot detail, the straightforward idea is that we have the right to divide both the numerator and also denominator of the portion 5/10 by 5 to uncover that it has an equivalent and simpler representation of 1/2. But don’t worry if that all sounds favor a bunch of crazy talk appropriate now—we’ll look in ~ it in more detail quickly enough.

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How to transform terminating decimal to fractions

Okay, we’re currently ready to relocate on to a more complex problem. Let’s convert a end decimal choose 0.875 into a fraction. As we talked around in the earlier short article on decimals, the 8 in 0.875 to represent 8/10, the 7 to represent 7/100, and the 5 to represent 5/1000. So the decimal number 0.875 is equal to8/10 + 7/100 + 5/1000But rather of worrying around how to actually add up every one of these fractions (which is an additional topic the we’ll talk around in a future article), we deserve to simplify things by very first writing 0.875 as0.875 = 0.800 + 0.070 + 0.005When we do this, you deserve to see the 0.875 is same to the sum of 800/1000 + 70/1000 + 5/1000 because that a grand full of (800 + 70 + 5)/1000 = 875/1000. And also that’s the answer!As through our earlier problem, it turns out that we deserve to reduce this fraction to the lowest state by dividing its numerator and denominator by 125. Law so, we uncover that 0.875 = 875/1000 is identical to 7/8. But, again, don’t problem if you don’t understand how reducing to lowest terms functions for now…we’ll come ago and talk about that in more detail soon.

Practice problems

So that’s how you transform a end decimal into an indistinguishable fraction. How around a repeating decimal number such together 0.333… or 0.818181…? Well, unfortunately, we’re out of time because that today. Which way that we’ll handle repeating decimals next time. However to make sure you’re increase to speed with convert terminating decimals, below are a couple of practice difficulties for you to try.1.4=______0.125=______0.800=______You can discover the answer below, but shot it there is no peeking, first!

Practice difficulty answers

1.4=1 and 4/10. We have the right to reduce this to lowest terms by dividing the numerator and denominator or 4/10 by 2 to gain the equivalent fraction 1 and 2/5.0.125=125/1000. We can reduce this come lowest state by splitting the numerator and also denominator through 125 to gain the equivalent portion 1/8.0.800=8/10. Once again, we can reduce this to lowest terms by separating the top and bottom by 2 to acquire the equivalent portion 4/5. The zeros on the finish of 0.800 don’t adjust anything about the problem since they simply tell us that there room zero hundredths and zero thousandths!